8 supplement to chapter -...

SUPPLEMENT OUTLINEIntroduction, 8S-2

Location Decisions, 8S-3

Other Applications, 8S-3

Computer Solutions, 8S-4

Problem, 8S-5

Reading: Some Applications of theTransportation Model, 8S-6

Mini-Case: Nu-Kote International, 8S-7

Bibliography and Further Reading, 8S-7

The Transportation Model

LEARNING OBJECTIVES

After completing this supplement,you should be able to:

1 Describe the nature of atransportation problem.

2 Set up transportationproblems as a transportationmodel.

3 Solve a transportation model.

8S-1

SUPPLEMENT TO

CHAPTER8

ste51675_ch08_suppl.qxd 10/11/2006 1:47 PM Page 8S-1

INTRODUCTIONThe transportation problem involves finding the lowest-cost plan for distributing goods or supplies from multiple origins to multiple destinations that demand the goods. Forinstance, a firm might have three factories, all of which are capable of producing identi-cal units of the same product, and four warehouses that stock or demand those products,as depicted in Figure 8S–1. The transportation model can be used to represent the struc-ture and data of transportation problem and to solve it to determine how to allocate thesupplies available from the various factories to the warehouses that stock or demand thosegoods, in such a way that total shipping cost is minimized. Usually, analysis of the problemwill produce a shipping plan that pertains to a certain period of time (day, week), althoughonce the plan is established it will generally not change unless one or more of the para-meters of the problem (supply, demand, unit shipping cost) changes.

Although Figure 8S–1 illustrates the nature of the transportation problem, in real lifemanagers must often deal with allocation problems that are considerably larger in scope.A beer maker may have four or five breweries and hundreds or even thousands ofdistributors, and an automobile manufacturer may have eight assembly plants scatteredthroughout the United States and Canada and thousands of dealers that must be suppliedwith those cars. In such cases, the ability to identify the optimal distribution plan makesthe transportation model very important.

The shipping (supply) points can be factories, warehouses, or any other place fromwhich goods are sent. Destinations can be warehouses, retail stores, or other points thatreceive goods. The information needed to use the model consists of the following:

1. A list of the origins and each one’s capacity or supply quantity per period.

2. A list of the destinations and each one’s demand per period.

3. The unit cost of shipping items from each origin to each destination.

This information is arranged into a transportation table (see Table 8S–1).The output from the transportation model is the optimal quantity to be transported on

each supplier-demand point route and may be displayed inside the cells of the transportationtable.

8S-2 SUPPLEMENT TO CHAPTER EIGHT THE TRANSPORTATION MODEL

D

(demand)

D

(demand)

D

(demand)

D

(demand)

S

(supply)

S

(supply)

S

(supply)

FIGURE 8S–1

The transportation probleminvolves determining aminimum-cost plan forshipping from multiple sourcesto multiple destinations


Use of the transportation model implies that certain assumptions are satisfied. Themajor ones are:

1. The items to be shipped are homogeneous (i.e., they are the same regardless of theirsource or destination).

2. Shipping cost per unit is the same regardless of the number of units shipped.

3. There is only one route or mode of transportation being used between each origin andeach destination.

LOCATION DECISIONSThe transportation model can be used to compare location alternatives in terms of theirimpact on the total distribution costs. The procedure involves working through aseparate problem for each location being considered and then comparing the resultingtotal costs.

If other costs, such as production costs, differ among locations, these can easily beincluded in the analysis, provided they can be determined on a per-unit basis. For eachfactory, just add the unit production cost to the unit transportation cost of factory-demandpoint pairs.

OTHER APPLICATIONSSome of the other uses of the model include aggregate operations planning (see Chapter 12),problems involving assignment of personnel or jobs to certain departments or machines,capacity planning, and trans-shipment problems.1

The use of the transportation model for capacity planning parallels its use for locationdecisions. An organization can subject proposed capacity alternatives to transportationanalysis to determine which one would generate the lowest total shipping cost. Forexample, a factory or warehouse that is closer to the market should have a larger capacitythan other locations.

SUPPLEMENT TO CHAPTER EIGHT THE TRANSPORTATION MODEL 8S-3

Cost to shipone unit fromfactory 1 towarehouse A.

Factory 1can supply 100units per period

Total supplycapacity perperiod

Total demandper period

Warehouse Bcan use 90 units per period

Warehouse

Factory

11774

88312

516108

A B C D

2

3

80 90 120 160 450450

150

200

100

Demand

Supply

TABLE 8S–1

A transportation table

1Trans-shipment relates to problems with major distribution centres that in turn redistribute to smaller marketdestinations. See, for example, W. J. Stevenson, Introduction to Management Science, 3rd ed. (Burr Ridge,IL: Richard D. Irwin, 1998).


COMPUTER SOLUTIONSAlthough manual solution of transportation model is fairly straightforward, computersolutions are generally preferred, particularly for moderate or large models. Many softwarepackages call for data input as in Table 8S–1.

Transportation model is a special case of a linear programming model. The decisionvariables for a transportation model are the quantities to be shipped. We can use thesymbol x1A to represent the decision variable for cell 1-A, x1B for cell 1-B, and so on. Theobjective function consists of the sum of cell costs times decision variables:

Minimize 4x1A � 7x1B � 7x1C � 1x1D � 12x2A � 3x2B � 8x2C

� 8x2D � 8x3A � 10x3B � 16x3C � 5x3D

In most cases, the capacity of each factory cannot be exceeded, but the demand at eachdestination must be met. Thus, we have

Supply (rows) x1A � x1B � x1C � x1D � 100

x2A � x2B � x2C � x2D � 200

x3A � x3B � x3C � x3D � 150

Demand (columns) x1A � x2A � x3A � 80

x1B � x2B � x3B � 90

x1C � x2C � x3C � 120

x1D � x2D � x3D � 160

If total supply is less than total demand, the problem will be infeasible. To avoid this,add an extra (dummy) row with the necessary supply and unit costs of zero to restorefeasibility.

Figure 8S–2 illustrates the Excel worksheet for the preceding problem. The top tableshows the unit costs, supplies and demands, whereas the bottom table shows the value ofdecision variables.


FIGURE 8S–2

Excel template for Table 8S–1


1. Solve this linear programming problem using the transportation method. Find the optimaltransportation plan and the minimum cost.

Minimize 8x11 � 2x12 � 5x13 � 2x21 � x22

� 3x23 � 7x31 � 2x32 � 6x33

Subject to x11 � x12 � x13 � 90x21 � x22 � x23 � 105x31 � x32 � x33 � 105x11 � x21 � x31 � 150x12 � x22 � x32 � 75x13 � x23 � x33 � 75All variables � 0

2. A toy manufacturer wants to open a third warehouse that will supply three retail outlets. Thenew warehouse can supply 500 units of backyard playsets per week. Two locations are beingstudied, N1 and N2. Transportation costs for location N1 to stores A, B, and C are $6, $8, and$7, respectively; for location N2, the costs are $10, $6, and $4, respectively. The existingsystem’s data are shown in the following table. Which location would result in the lowertransportation costs for the system?

3. A large firm is contemplating construction of a new manufacturing facility. The two leadinglocations are Hamilton and Thunder Bay. The new factory would have a supply capacity of 160 units per week. Transportation costs from each potential location and existing manufac-turing facilities locations 1, 2, and 3 to markets A, B, and C are shown in the following tables.Determine which new location would provide lower total transportation cost.

From From Hamilton Cost Thunder Bay Cost to per Unit to per Unit

A . . . . . . $18 A . . . . . . . . . $7B . . . . . . 8 B . . . . . . . . . 17C . . . . . . 13 C . . . . . . . . . 13

4. A large retailer is planning to open a new store (the retailer currently has stores A and B). Threelocations are currently under consideration: South Coast Plaza (SCP), Fashion Island (FI), andLaguna Hills (LH). Transportation costs from the warehouses 1, 2, and 3 to the new locationsand transportation costs, demands, and supplies from the warehouses to the existing locations

220220220

140

150

210

Demand(units/week)

A B C

1

2

3

Supply(units/week)

10 14 10

20

1211

17

11

12

1

A B C

2

600400 350

400

500

Demand(units/week)

Capacity(units/week)To:

From:

Warehouse

Store

8 3

5 9

7

10


PROBLEMS


are shown below. Each of the new store locations has a demand potential of 300 units per week.Which new store location would yield the lowest transportation costs for the system?

FROM To

WAREHOUSE SCP FI LH

1 $4 $7 $52 11 6 53 5 5 6

9

7

14 18

15

10

500

660

340

200

400Demand(units/week)

Supply(units/week)A B

1

2

3


Simmons.* In the early 1980s, Simmons Co., the largemattress and box spring maker, had 19 plants and 67 ware-houses in the U.S. The large numbers of facilities had resultedin excess inventories and inefficiencies. Then the companystarted to close warehouses and produce to order. The produc-tion lead time was reduced to 4 to 5 days. Simmons usedOptiSite software of MicroAnalytics to decide which plants tokeep open and which market to serve from each plant. OptiSiteused the transportation model and solution technique. Toyota/Lexus Service Parts Distribution Network in the

U.S.** At the end of 1990s, Toyota decided to reevaluate itsU.S. service parts distribution network which was 30 yearsold. Toyota used two major distribution centres in Californiaand Kentucky, with additional regional distribution centres in Los Angeles, San Francisco, Portland, Kansas City, New York, Cincinnati, Baltimore, Chicago, and Boston, witha Lexus-only centre in Jacksonville, Florida. Toyota usedInsight’s Sails software to determine the optimality of itsdistribution network. Sails uses a network model similar to thetransportation model. Insight determined that the capacity of the Kansas City centre was limited and proposed opening anew Lexus-only centre in Texas. Toyota agreed.

R E A D I N G

Some Applications ofthe TransportationModel

*www.bestroutes.com/testimonials.html#simmons, reprinted from R. Bowman, “Keying into Versatility,” Distribution 92(5), May 1993, pp. 54–61.**www.bestroutes.com/pdf/simmons.pdf


Bierman, Harold, Charles P. Bonini, and Warren H.Hausman. Quantitative Analysis for Business Deci-sions. 8th ed. Burr Ridge, IL: Richard D. Irwin, 1991.

Stevenson, William J. Introduction to ManagementScience. 3rd ed. Burr Ridge, IL: Richard D. Irwin,1998.


Nu-Kote is a manufacturer and distributor of inkjet andlaser inks and cartridges. Nu-Kote has a plant and ware-

house in Franklin (Tennessee), and plants in China,Chatsworth (California), Rochester (N.Y.), and Connellsville(Pennsylvania). Approximately 80 percent of finished goods aremade in China (shipped via Chatsworth plant to Franklin) andin the Chatsworth plant. However, approximately 80 percentof customer demand is in the east coast. Nu-Kote transports allfinished goods to the Franklin warehouse (using full truckloads) and from there distributes them to customers using less-than truck loads. Management is interested to know if a secondwarehouse (e.g., in Chatsworth) would significantly reduce thetotal transportation cost. There are hundreds of customers, butfor simplicity we assume that there are only two customers:Cust East with demand of 2,000 units per year (each unit is

1,000 cases), and Cust West with demand of 500 units per year.Suppose that the supply from the Chatsworth plant (includingproducts made in China) is 2,000 units per year and fromFranklin is 500 units per year. Ignore the other plants. Supposethat the unit transportation cost from the Chatsworth ware-house (if built) to Cust West is $200 per unit and to Cust Eastis $1,100 per unit. Also, suppose that the transportation costfrom Franklin Warehouse to Cust West is $1,000 per unit andto Cust East is $300 per unit. The transportation cost fromChatsworth plant to Chatsworth warehouse and from Franklinplant to Franklin warehouse is negligible. The transportationcost between the two warehouses is $700 per unit. All otherroutes are infeasible. Formulate this problem as a transporta-tion model and solve it using Excel. Is the optimal solutionwhat you intuitively expected?

Source: L. J. LeBlanc et al., “Nu-Kote’s Spreadsheet Linear-Programming Models for Optimizing Transportation,” Interfaces34(2), Mar/Apr 2004, pp. 139–146.

M I N I - C A S E

Nu-Kote International

BIBLIOGRAPHY ANDFURTHER READING

Teknaprint Systems.4 Teknaprint manufactures imaging andprinting equipment. In the late 1990s, Teknaprint wanted toreduce its logistics costs. It had two distribution centres (DC)in Irvine, California, and Richmond, Virginia. The Irvine DCwas used to handle products of the Mexican plant and theRichmond DC was used for the Richmond plant. From the

two DCs, products were transported to more than 11,000customers. There were many duplications of products andmany markets were served by both DCs. Teknaprint usedLogicTool’s LogicNet to model its current distribution system(the Baseline) and to provide an optimal distribution network.

4www.logic-tools.com/customers/case_study_teknaprint.pdf


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